« PreviousContinue »
(2) y2 = 2 cx + c = –22; so that c=-X; consequently (2) is a singular solution.
In the case however where the general integral is not known, an inversion of the process of Articles 405 and (106) will determine whether y = F (x) is a particular integral or a singular solution; that is, by inquiring whether - Sy is not or is rendered infinite by the substitution of f(x) for y; or whether a vanishes by the same substitution. Of this process we subjoin some examples, and shall first take that which has just been considered.
Ex. 1. The equation yy'2 + 2 xy' - y=f(x, y, y) = 0 is satisfied by (1) yo = 2x+1; (2) by y2 + x4 = 0; are they singular solutions or particular integrals ?
CO) = 2yy’+2x, which does not vanish for the relation (1), but does vanish for (2): therefore (1) is a particular integral, and (2) is a singular solution.
Ex. 2. The equation y? + Y ý' + x = 0) is satisfied when ya +(2— 1)= 0; is this expression a singular solution or a particular integral ?
= 2y' + y,
I dyl and as this does not vanish when yo + (x — 1)2 = 0, this function is a particular integral.
Ex. 3. y = ax satisfies the equation (1-xo)y' + xy-a = 0; prove that this is a particular integral.
Through the preceding Articles we have considered the differential equation to be a function of x, y, uy, and have deduced
F'dx' our results on this supposition; we might however just as well have considered it to be a function of y, x, , in which case the conditions for a singular solution would be
**dy and the resulting equation x = F(y) must of course satisfy the differential equation.
408.] Thus far we have investigated singular solutions with reference to the differential equations of which they are solutions; we proceed now to deduce them from, and to point out their relation to, the general integrals of the differential equations; and herein we shall recur to the characteristic property of them; viz., that they are particular forms of the general integral when the arbitrary constant of integration is replaced by a function of the variables, whereby the solution becomes a function of x and y only, and also is such as to satisfy the differential equation. Let the general integral of a differential equation be F(x, y, c) = 0,
(206) where c is an arbitrary constant introduced in integration; then the theory of the formation of differential equations shews that the given differential equation has been formed by the elimination of c between (206) and
(as) dx + () dy = 0. (207)
Now it is to be considered whether the same valu not be obtained from an equation of the form (206), if c is replaced by a function of x and y, say of the form $ (x, y), which we shall abbreviate into o for convenience of notation; because if this is possible, the function hereby obtained is a singular solution. Suppose then the integral to be F(x, y, p) = 0;
(208) - volante de + (dy) dy + (1) lip = 0; (209) since however o denotes a function of x and y,
dop = (het) dx + (49) dy ; and substituting this in (208), we have
This will be identical with (207), if
Now this condition can be satisfied in three different ways;
Firstly, if (cho) = ( )= 0; which shew that © contains neither x nor y; that is, o = a constant; and thus we have the general integral if the constant is arbitrary, and a particular integral if it has a particular value.
409.] Secondly, (211) is satisfied if () =0; and if ® is eliminated by means of this condition and of (208), or, what amounts to the same thing, if we eliminate c between F(x, y, c) = 0, and
(212) the resulting equation will be a relation between x and y which will satisfy the given differential equation; and will not be a particular integral, unless c should happen to be equal to a particular constant previously involved in the differential equation.
The following are examples of this theorem.
Ex. 1. The general integral of a differential equation is y=c(x+c)2; it is required to find the singular solution.
F(x, y, c) = y-C(x + c)2 = 0; .. (.)= -(x+ c) (x +3c) = 0;
.. 2 = -; X=-30; of which values the former makes y = 0, and as the same result is obtained if c = 0, it gives a particular integral. The second value gives
4x3 + 27 y = 0, which is the singular solution.
Ex. 2. The general integral of a differential equation is cx— cy+a=0; it is required to find the singular solution.
F(X, Y, C) = cx- cy+a = 0; : :: (F) = 2cx—y = 0, if c=
whence y2 = 4ax; and as no particular value of the constant can give this equation, it is a singular solution.
Ex. 3. The general integral of a differential equation is y-cx—(52 + a2c2)*=0; it is required to find the singular solution.
F(x, y, c) = y-CX—(62 + a2c2)t = 0;
alue givetves a part
.. (F)=-~-_ ac
a(a? — Q2) whence we have + = 1; and this is the singular solution.
The process by which singular solutions are thus derived from the general integral is evidently identical with that by which the envelope of a family of lines, each individual of which is given by a particular value of an arbitrary constant, has been determined in Section 2, Chapter XIII, Vol. I. Thus the general integral involving the arbitrary constant represents the family of lines; the particular integral, a particular value having been given to the arbitrary constant, expresses an individual of the family; and the singular solution which is determined by the elimination of the constant between the general integral and its c-derived function, expresses the envelope of these particular lines. In illustration of these remarks let us take the preceding Ex. 2. The general integral therein given is the general integral
de les + a = 0; and the integral may be expressed in the form
y = ca This is manifestly the equation to a straight line; and to a series of straight lines, if c is considered a variable parameter; and the envelope of all these is the singular solution, and is a parabola whose latus rectum is 4a, as appears from the preceding example; it will be at once seen that the equation is that to the tangent of a parabola in what is sometimes called the magical form.
The comparison of the preceding differential equation and its general integral shews that ay in the first is replaced in the second by c: the c-differentiation therefore of the second produces the same result in terms of c as the am - differentiation of the first produces for Hence this method of deducing the singular solution is, in this form of equation at least, the same as that investigated and applied in Art. 406 : we have not therefore hereby obtained a more general method.
410.] Thirdly, (211) will be satisfied if simultaneously we have
Come = (49) = 20; (213) which condition is the same as that determined in Art. 405. In respect of this circumstance let it be observed that it is inconsistent with the very first principles of differentiation that the derived-functions should have infinite values : if they have, the rules according to which they have been found fail. Now in differentiating a function of one variable only, say of x, it may be that its derived-function becomes infinite for a particular constant value of the variable: thus, for instance, if y = (x2 – a?), y'= 0, if x = + a; but in a function of two variables, as, for instance, u = (x2+y* — a?)} = 0, () = co, and () = co, if x2 + y2 = a?: that is, the total differential of u is infinite for this relation between x and y. Here then we have met with a case which is beside the common rules of differentiation; but which is of great importance in reference to singular solutions. For suppose u to involve other functions of x and y which are not infinite for the particular relation which makes the above values infinite, and suppose it to contain a function of an arbitrary constant, and the derived-function of it with reference to this arbitrary constant not to become infinite for this relation between x and y, then all these quantities must be neglected in comparison with those which become infinite ; and therefore the function of x and y which renders them infinite satisfies the differential equation, and is independent of the arbitrary constant which the general integral contains: and as this last property is characteristic of a singular solution, the function which satisfies these conditions is a singular solution. Hence we infer that if a function of x and y, which is independent of the arbitrary constant of integration, renders infinite Gon) and ..), and at the same time satisfies the differential equation, it is a singular solution, provided that it cannot be obtained by giving any particular constant value to the general constant of integration.
With regard to this criterion of a singular solution it must be observed that (?) which is the same as () must not become infinite simultaneously with () and (); for if this is the case, (211) takes an indeterminate form.